Motivated by recent Transition Region and Coronal Explorer (TRACE) observations of damped oscillations in coronal loops, we consider analytically the motion of an inhomogeneous coronal magnetic tube of radius a in a zero-plasma. An initially perturbed tube may vibrate in its kink mode of oscillation, but those vibrations are damped. The damping is due to resonant absorption, acting in the inhomogeneous regions of the tube, which leads to a transfer of energy from the kink mode to Alfvén (azimuthal) oscillations within the inhomogeneous layer. We determine explicitly the decrement (decay time À1 ) for a coronal flux tube whose plasma density varies only in a thin layer of thickness ' on the tube boundary. The effect of viscosity is also considered. We show that, in general, the problem involves two distinct timescales, À1 and ! À1 k R 1=3 , where R is the Reynolds number and ! k is the frequency of the kink mode. Under coronal conditions (when À1 5 ! À1 k R 1=3 ), the characteristic damping time of global oscillations is À1 . During this time, most of the energy in the initial perturbation is transferred into a resonant absorption layer of thickness of order ' 2 =a, with motions in this layer having an amplitude of order a=' times the initial amplitude. We apply our results to the observations, suggesting that loop oscillations decay principally because of inhomogeneities in the loop. Our theory suggests that only those loops with density inhomogeneities on a small scale (confined to within a thin layer of order a=! k in thickness) are able to support coherent oscillations for any length of time and so be observable. Loops with a more gradual density variation, on the scale of the tube radius a, do not exhibit pronounced oscillations.

The details of the mechanism(s) responsible for the observed heating and dynamics of the solar atmosphere still remain a mystery. Magnetohydrodynamic waves are thought to have a vital role in this process. Although it has been shown that incompressible waves are ubiquitous in off-limb solar atmospheric observations, their energy cannot be readily dissipated. Here we provide, for the first time, on-disk observation and identification of concurrent magnetohydrodynamic wave modes, both compressible and incompressible, in the solar chromosphere. The observed ubiquity and estimated energy flux associated with the detected magnetohydrodynamic waves suggest the chromosphere is a vast reservoir of wave energy with the potential to meet chromospheric and coronal heating requirements. We are also able to propose an upper bound on the flux of the observed wave energy that is able to reach the corona based on observational constraints, which has important implications for the suggested mechanism(s) for quiescent coronal heating.

The linear theory of MHD resonant waves in inhomogeneous plasmas is reviewed. The review starts from discussing the properties of driven resonant MHD waves. The dissipative solutions in Alfvén and slow dissipative layers are presented. The important concept of connection formulae is introduced. Next, we proceed on to non-stationary resonant MHD waves. The relation between quasi-modes of ideal MHD and eigenmodes of dissipative MHD are discussed. The solution describing the wave motion in non-stationary dissipative layers is given. It is shown that the connection formulae remain valid for nonstationary resonant MHD waves. The initial-value problem for resonant MHD waves is considered. The application of theory of resonant MHD waves to solar physics is discussed.

On 14 July 1998 TRACE observed transverse oscillations of a coronal loop generated by an external disturbance most probable caused by a solar flare. These oscillations were interpreted as standing fast kink waves in a magnetic flux tube. Firstly, in this review we embark on the discussion of the theory of waves and oscillations in a homogeneous straight magnetic cylinder with the particular emphasis on fast kink waves. Next, we consider the effects of stratification, loop expansion, loop curvature, non-circular cross-section, loop shape and magnetic twist.An important property of observed transverse coronal loop oscillations is their fast damping. We briefly review the different mechanisms suggested for explaining the rapid damping phenomenon. After that we concentrate on damping due to resonant absorption. We describe the latest analytical results obtained with the use of thin transitional layer approximation, and then compare these results with numerical findings obtained for arbitrary density variation inside the flux tube.Very often collective oscillations of an array of coronal magnetic loops are observed. It is natural to start studying this phenomenon from the system of two coronal loops. We describe very recent analytical and numerical results of studying collective oscillations of two parallel homogeneous coronal loops.The implication of the theoretical results for coronal seismology is briefly discussed. We describe the estimates of magnetic field magnitude obtained from the observed fundamental frequency of oscillations, and the estimates of the coronal scale height obtained using the simultaneous observations of the fundamental frequency and the frequency of the first overtone of kink oscillations.In the last part of the review we summarise the most outstanding and acute problems in the theory of the coronal loop transverse oscillations.

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